We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number
md5=a9723460f0068d8462ee8bf21db4627e" title="Click to view the MathML source">α∈(0,1) a commutative nonassociative algebra
md5=bed1c8a4e6abb16aba956b1cc8e5e179" title="Click to view the MathML source">Aα whose codimension sequence
md5=22198f13359863c0e17a47eae2fb8c5f" title="Click to view the MathML source">cn(Aα),
md5=2b1615f9106fd1ef447d3f3c320c6369" title="Click to view the MathML source">n=1,2,… , is polynomially bounded and
md5=016092d556caa3d59a2fef9f7e2f7d61" title="Click to view the MathML source">limlogncn(Aα)=3+α.
As an application we are able to construct a new example of a variety with an infinite basis of identities.